Question: Isabella is playing with her yo-yo. The vertical distance $Y$ (in $\text{cm}$ ) between the yo-yo and her hand $t$ seconds after she first spins it out is modeled by the following function. Here, $t$ is entered in radians. $Y(t) = {40}\cos\left({\dfrac{2\pi}{3}}t\right) - {71}$ How long does it take the yo-yo to fall all the way down from its peak, and then rise up to a vertical distance of $-80\text{ cm}$ ? Round your final answer to the nearest tenth of a second.
Answer: Converting the problem into mathematical terms $Y(t) = {40}\cos\left({{\dfrac{2\pi}{3}}}t\right) - {71}$ has a period of $\dfrac{2\pi}{{\scriptsize\dfrac{2\pi}{3}}}=3$ seconds. When the yo-yo falls all the way from its peak, it already passes the $-80\text{ cm}$ mark once. Therefore, we want to find the second solution to the equation $Y(t)=-80$ within the period $0<t<3$. The answer The equation's two solutions within the desired period (rounded to the nearest tenth of a second) are $0.9$ and $2.1$. Therefore, it takes about $2.1$ seconds for the yo-yo to fall down and then up to $-80\text{ cm}$.